In the past few years, many researchers have studied elastic optical networks, which exhibit a dramatically improved transmission capacity compared with conventional optical networks. However, the transmission capacity per fiber will soon reach the physical limit for traditional single-mode fibers. Multicore fiber (MCF) is among the innovative fibers based on space-division multiplexing technology. MCF has multiple cores and achieves a far larger transmission capacity than traditional single-mode fibers. However, signals transmitted in these crowded multiple cores interfere with each other and are degraded. This degradation has a serious impact on the network resource management in elastic optical networks. Previous research indicates that the crosstalk effect of MCF is dependent on an arrangement of signals regarding the spectrum and core. Therefore, we approach the problem of the crosstalk in MCF with regard to the spectrum and core allocation, from the network perspective. First, this paper summarizes the related work regarding elastic optical networks and MCFs. Next, we propose an “on-demand” spectrum and core allocation method that reduces both the crosstalk and fragmentation in elastic optical networks with MCFs. This proposed method is based on two predefined policies related to the crosstalk and fragmentation. The first, the core prioritization policy, is based on the MCF’s structure, and the other is a core classification policy based on the required bandwidth of the connections. The core prioritization policy realizes the core allocation that reduces crosstalk by avoiding filling adjacent cores. The core classification policy reduces the spectrum fragmentation by allocating a uniform bandwidth connection for each core. Finally, we evaluate the proposed method using computer simulations. The results indicate that the proposed method can, under various network conditions, improve both the crosstalk and blocking probability of the total network through our two policies.